QB 
4-4 


IC-NRLF 


SUGGESTIONS  TO  TEACHERS 


DESIGNED    TO    ACCOMPANY 


A  TEXT-BOOK   OF   ASTRONOMY 


BY 


GEORGE   C.   COMSTOCK 

DIRECTOR   OF   THE   WASHBURN   OBSERVATORY   AND 

PROFESSOR    OF    ASTRONOMY    IN    THE 

UNIVERSITY  OF    WISCONSIN 


OF  THE 

UNIVERSITY 


NEW    YORK 

D.    APPLETON   AND   COMPANY 
1901 


TWENTIETH   CENTURY  TEXT-BOOKS 


SUGGESTIONS  TO  TEACHERS 

DESIGNED   TO    ACCOMPANY 

A  TEXT-BOOK   OF   ASTRONOMY 


BY 

GEORGE  C  COMSTOCK 

DIRECTOR    OF   THE   WASHBURN    OBSERVATORY    AND 

PROFESSOR   OF    ASTRONOMY    IN    THE 

UNIVERSITY    OF    WISCONSIN 


NEW    YORK 

D.    APPLETON   AND   COMPANY 
1901 


COPYRIGHT,  1901 
BY   D.   APPLETON   AND   COMPANY 


ASTRONOMY 


SUGGESTIONS   TO   TEACHERS 

THE  suggestions  contained  in  the  following  pages  are 
intended  primarily  for  the  inexperienced  teacher  who  is 
endeavoring  for  the  first  time  to  use  the  author's  Text-Book 
of  Astronomy.  The  experienced  teacher  may  be  left  to 
follow  his  own  methods,  but  the  beginner  is  advised  at  the 
outset  to  read  carefully  the  whole  Text-Book  before  attempt- 
ing to  teach  any  part  of  it. 

Cultivate  your  own  interest  in  and  knowledge  of  the 
subject  by  reading  also  other  works — e.  g.,  at  least  a  por- 
tion of  those  named  in  the  Bibliography  at  the  end  of  the 
Text- Book.  Your  function  as  a  teacher  is  to  awaken  in 
your  pupils  an  interest  in  the  subject  quite  as  much  as  to 
give  them  instruction  in  it,  and  neither  of  these  purposes 
will  be  satisfactorily  accomplished  without  enthusiasm  on 
your  own  part  and  a  wider  information  than  is  contained 
in  any  one  elementary  text. 

In  your  own  reading  of  the  text,  note  how  the  point  of 
view  changes  as  you  pass  from  beginning  to  end.  The 
initial  chapters  are  devoted  to  things  in  the  sky  which  can 
be  seen  with  the  unaided  vision,  and  which  the  pupil  should 
be  taught  to  see  for  himself.  Then  follows  reasoning  about 
these  things,  using  in  some  degree  elementary  mathemat- 
ical methods,  much  as  the  science  was  developed  among 
primitive  people  some  thousands  of  years  ago.  Then  fol- 
lows in  Chapters  IV  to  VII  an  elementary  statement  of 

~84374 


2  ASTRONOMY 

some  of  the  results  attained  by  a  more  profound  study  of 
the  phenomena  which  the  pupil  has  learned  to  observe.  But 
modern  science  deals  with  much  more  complicated  data 
than  have  been  presented  thus  far,  and  Chapter  VIII,  upon 
instruments,  shows  how  the  senses  may  be  helped  to  obtain 
from  the  stars  impressions  and  information  that  are  quite 
beyond  the  reach  of  unaided  vision.  Subsequent  chapters  of 
the  book  are  devoted  mainly  to  a  description  of  the  heavenly 
bodies  as  they  are  made  known  to  us  by  the  new  sources  of 
information. 

An  attempt  has  been  made  to  bring  before  the  pupil,  by 
means  of  illustrations,  the  kind  of  information  and  evidence 
which  the  astronomer  obtains  at  first  hand  from  the  tele- 
scope, camera,  and  spectroscope,  and  to  draw  from  these 
illustrations  the  conclusions  presented  in  the  text.  The 
teacher's  attention  is  especially  invited  to  this  feature  of 
the  book  which,  if  properly  used,  may  be  made  to  add 
greatly  to..tjhe  educational  value  of  the  work.  Seek  to  train 
the  pupil's  perceptions  and  reasoning  faculty  as  well  as  his 
memory.  Beware  of  putting  too  much  stress  upon  learn- 
ing facts  as  the  substance  of  work  in  science.  Although  a 
certain  basis  of  fact  is  essential,  its  acquisition  from  a  book 
has  little  educational  value,  and  emphasis  ought  not  to  be 
placed  here,  but  upon  the  relation  of  facts  to  each  other, 
and  upon  the  ability  to  combine  and  make  use  of  them. 

With  this  end  in  view  numerous  exercises  and  problems 
have  been  introduced  into  the  text,  and  many  facts  inter- 
esting and  important  to  a  complete  view  of  the  science  of 
astronomy  have  been  omitted  in  order  to  make  room  for 
such  work. 

Some  of  the  problems  involve  numerical  calculations 
that  would  be  tedious  if  carried  out  by  the  ordinary  arith- 
metical processes,  but  which  are  made  short  and  simple  by 
the  use  of  logarithms.  To  facilitate  this  work  a  table  of 
three-figure  logarithms  is  given  at  the  end  of  this  manual. 
For  convenient  use  in  the  class  room  it  may  be  removed 


SUGGESTIONS  TO  TEACHERS  3 

from  the  book  and  mounted  upon  a  piece  of  cardboard. 
Show  it  to  the  pupils,  and  without  going  far  into  the  the- 
ory of  logarithms,  explain  its  use  in  problems  of  multi- 
plication, division,  and  the  extraction  of  roots.  Be  espe- 
cially careful  in  your  explanations  of  interpolation  and  the 
process  of  finding  the  number  corresponding  to  a  given 
logarithm. 

The  out-of-door  work  to  be  done  by  pupils  in  connec- 
tion with  their  study  of  astronomy  depends  so  much  upon 
conditions  of  climate  and  weather  that  few  precepts  of 
general  application  can  be  laid  down  regarding  it,  but  the 
pupil  should  be  taught  to  look  at  the  stars  whenever  he  is 
out  of  doors  on  a  clear  evening.  Much  may  be  accom- 
plished in  this  way  without  appreciable  expenditure  of 
time.  Take  advantage  of  the  fact  that  the  subject-matter 
of  astronomy  lies  at  your  own  door  and  require  -  no  expe- 
dition to  the  country  in  order  to  find  it. 

In  addition  to  the  suggested  observation  of  the  sky,  the 
text  contains  a  considerable  amount  of  pure1^  laboratory 
work  with  protractor  and  scale,  in  the  measurement  of  pho- 
tographs, and  in  the  graphical  treatment  of  data.  This  is 
available  at  all  times,  and  should  be  made  prominent  in  the 
pupil's  study.  Assign  specific  problems  of  this  kind  to  be 
worked  out  in  good  form,  entered  in  a  note  book,  and  pre- 
served as  a  part  of  the  pupil's  work.  It  will  of  course  be 
expedient  to  give  different  pupils  slightly  different  data  in 
order  to  avoid  the  temptation  to  collusion  in  their  work. 
Make  the  note  book  count  as  a  part  of  your  final  exami- 
nation. 

Before  beginning  this  work,  examine  each  pupil's  pro- 
tractor and,  if  necessary,  trim  it  with  a  pair  of  shears  ex- 
actly to  the  line  printed  along  the  straight  edge.  Neglect 
of  this  precaution  is  sure  to  result  in  bad  work — e.  g.,  in 
Exercise  2, 


ASTRONOMY 


CHAPTER   I 

1-3.  Do  not  be  satisfied  with  one  lesson  on  this  chapter. 
Come  back  to  it  once  or  twice  a  week,  and  repeat  the  exer- 
cises until  the  pupils  acquire  considerable  facility  and  some 
pride  in  doing  their  work  with  precision. 

Exercises  4  and  5  may  be  practiced  in  a  window  facing 
south.  If  possible  have  the  window  open,  so  that  the  glass 
shall  not  interfere  with  the  sun's  rays.  With  reasonable 
care  the  sun's  altitude  may  be  measured  in  this  way  to 
within  a  small  fraction  of  a  degree. 

5.  Point  out  to  the  pupil  that  in  Fig.  5  the  time  of  the 
sun's  greatest  altitude,  llh.  50m.,  comes  near  noon  but  not 
exactly  at  noon,  and  that  the  explanation  of  this  discrep- 
ancy is  to  be  found  in  the  equation  of  time,  §  53. 

6.  Solve  the  problem  at  the  end  of  this  section  by  meas- 
uring the  distance  of  the  curve  above  the  base  line  at  the 
point  corresponding  to  12h.   and  2.5h. — e.  g.,  30  millime- 
ters and  23.3  millimeters — and  using  the  proportion 

30.0  :  23.3  : :  57°  :  the  required  altitude. 

7.  This  section,  although  not  marked  as  an  .exercise, 
should  be  treated  as  one,  and  the  prescribed  operations  exe- 
cuted by  the   pupils.     The   method   here   outlined   is,  of 
course,  available  for  determining  the  diameter  of  any  heav- 
enly body  which  presents  a  sensible  disk,  but  can  not  be 
applied  to  the  earth. 

CHAPTER   II 

The  plumb-line  apparatus  which  is  introduced  in  this 
chapter  should  be  kept  standing  in  the  recitation  room, 
and  pupils  should  be  encouraged  to  use  it  at  odd  moments 
as  they  can  find  time.  Employ  it  in  connection  with  Ex- 
ercises 4  and  5  after  the  methods  of  Chapter  I  have  become 
familiar. 


SUGGESTIONS  TO  TEACHERS  5 

Before  using  the  apparatus  for  observing  stars,  the 
pupil  should  have  acquired  some  familiarity  with  the  prin- 
cipal constellations.  This  may  be  done  without  any  severe 
tax  upon  his  time  if  he  is  taught  to  watch  the  sky  when- 
ever he  is  out  of  doors  on  a  clear  evening.  The  habit,  if 
once  acquired,  will  be  a  lifelong  source  of  pleasure,  and  the 
teacher  who  does  not  inculcate  it  ignores  one  of  the  best 
elements  in  the  study  of  astronomy.  Study  the  star  maps. 

9.  The  star  maps  are  to  be  found  at  pages  124  and  190, 
instead  of  at  beginning  and  end  of  book. 

10.  Answers. — A  straight  line.      Because    they  point 
toward  Polaris.     The  letter  W. 

11.  The  rate  at  which  the  stars  turn  about  the  pole  is 
360°  in  23h.  56m.  3.5s. 

12.  An  ordinary  clock  may  be  turned  into  a  sidereal 
clock  by  moving  the  regulator  so  as  to  make  it  gain  3m. 
56.5s.  per  day. 

13.  Each  trail  subtends  at  the  center  of  the  picture  an 
angle  of  15°,  and  the  exposure  therefore  lasted  one  hour. 
360°  :  15°  : :  24h.  :  Ih. 

14.  15.  The  purpose  of  these  sections  is  to  connect  the 
circular  motion  of  stars  near  the  pole  with  the  diurnal  mo- 
tion of  sun  and  moon,  and  to  show  that  the  latter  motion 
is  like  that  of  the  stars,  save  that  it  takes  place  in  a  larger 
circle,  a  part  of  which  dips  below  the  horizon. 

The  Pleiades  are  too  far  from  the  north  pole  to  show  in 
Plate  I. 

16.  Pronounce,  Ant-ar-es. 

18.  Note  that  to  an  observer  at  the  north  pole,  Polaris 
would  stand  very  near  the  zenith.     As  the  observer  travels 
toward   the   south   the   star  will  apparently  move   down 
toward  the  north  horizon,  will  reach  the  horizon  when  the 
observer  reaches  the  equator,  and  will  become  invisible  as 
he  goes  farther  south.     Pronounce,  Gas-tor,  SpT-ca,  Al-tar. 

19.  The  meridian  line  may  be  equally  well  laid  out  at 
any  convenient  hour  when  Polaris  is  visible,  regardless  of 


6  ASTRONOMY 

its  place  in  the  diurnal  path  by  means  of  a  method  given 
in  the  periodical,  Popular  Astronomy,  Vol.  IX,  No.  5. 

20.  "  The  time  "  always  means  the  hour  angle  of  some 
selected  point  of  the  sky,  the  vernal  equinox  for  sidereal 
time  and  the  sun  for  solar  time. 

21.  It  is  well  to  put  considerable  emphasis  upon  the 
matter  of  exact  definitions.     If  possible  get  the  pupils  to 
criticise  the  definitions  of  the  text,  as  well  as  the  defini- 
tions offered  by  their  fellows. 

22.  Answers. — Right  ascension  and  declination  are  not 
changed  by  the  diurnal  motion.     The  hour  angle  of  any 
star  is  0  when  its  altitude  is  greatest,  and  the  same  is  very 
approximately  true  for  the  sun.     It  would  be  exactly  true 
if  there   were   no   progressive   change  in  its  declination. 
After  sunset  the  sun's   altitude  is  negative.     The  north 
pole  is   always  north  from  every  other  part  of  the   sky. 
The  intersection  points  are  north  and  south  in  the  case  of 
the  meridian,  east  and  west  in  case  of  the  equator. 


CHAPTER   III 

The  work  of  learning  the  principal  constellations,  which 
was  commenced  in  Chapter  II,  should  be  continued  through- 
out the  entire  period  given  to  the  study  of  astronomy,  and 
in  this  connection  the  construction  of  star  maps  will  be 
found  a  very  useful  exercise,  but  there  is  some  difficulty  in 
getting  it  well  done.  Select  for  a  beginning  some  constel- 
lation of  the  zodiac,  such  as  Taurus,  Leo,  or  Virgo,  and 
as  a  preparation  for  §§  28-31  have  the  pupil  learn  from  the 
sky  and  the  star  maps  the  sequence  in  which  the  twelve 
zodiacal  constellations  make  up  the  circuit  of  the  sky.  See 
Figs.  16  and  17.  Continue  this  work,  adding  one  new  con- 
stellation each  evening.  Note  carefully  the  relation  of 
each  new  constellation  to  the  old  ones. 

23.  Pronounce,  Al-deb-ar-an. 


SUGGESTIONS  TO  TEACHERS  f 

24.  The  line  joining  the  moon's  horns  is  nearly  perpen- 
dicular to  the  ecliptic.  The  greater  the  angular  distance 
of  the  moon  from  the  sun,  the  greater  is  its  visible  area  of 
illuminated  surface.  See  Chapter  IX  for  answers  to  other 
questions. 

27.  If  the  orbit  of  Mars  coincided  with  that  of  the 
earth,  the  planet  would  always  travel  along  the  ecliptic 
instead  of  moving  north  and  south  of  it,  as  shown  in 
Fig.  14. 

EXEKCISE  16. — Answers  are  to  be  obtained  by  direct 
measurement  from  Fig.  16.  For  the  date  of  opposition  of 
Jupiter,  note  from  the  figure  that  on  January  1,  1906,  the 
earth  is  not  on  line  between  the  sun  and  Jupiter,  nor  is  it 
on  line  at  the  dates  represented  by  II,  III,  etc.  At  XII  it 
is  nearly  on  line,  but  Jupiter  is  then  near  the  part  of  its 
orbit  marked  1907 ;  and  since  the  earth  will  require  several 
days  to  move  up  to  this  position,  the  planet's  opposition 
will  not  occur  until  near  the  end  of  December.  The  exact 
date  is  December  27th.  The  date  of  conjunction,  June 
llth,  may  be  determined  in  the  same  way. 

Mark  upon  Fig.  16,  for  any  given  date,  the  positions  of 
Jupiter  and  the  earth,  and  imagine  yourself  standing  upon 
the  position  of  the  earth  and  looking  toward  the  sun.  If 
Jupiter  appears  to  the  left  of  the  sun,  the  planet  will  be 
visible  in  the  evening  hours  ;  if  to  the  right  of  the  sun,  it 
may  be  seen  in  the  morning  sky.  The  same  method  may 
be  applied  to  any  other  planet. 

From  Fig.  16  the  constellation  opposite  to  the  sun  on 
January  1st  of  each  year  is  Gemini,  and  this  constellation 
will  then  reach  the  meridian  at  midnight,  and  so  will  every 
other  constellation  in  the  same  right  ascension — e.  g.,  Canis 
Major.  See  the  star  map. 

Note  that  the  margins  of  Figs.  16  and  17  show  the 
zodiacal  constellations,  and  not  the  so-called  signs  of  the 
zodiac. 

30,  31.  The  exercises  upon  Figs.  16  and  17  should  be 
2 


8  ASTRONOMY 

carefully  worked  out  with  reference  to  numerical  accuracy. 
The  parallel  lines  drawn  from  the  sun  are  required,  because 
the  figures  show  correctly  the  directions  of  the  constel- 
lations from  the  sun,  but  do  not  give  their  directions  cor- 
rectly for  any  other  point  in  the  figures.  With  reasonable 
care  the  positions  of  the  planets  in  the  sky  may  be  found  so 
closely  that  there  can  be  no  doubt  as  to  their  identification. 
Point  out  in  Fig.  17  how  far  the  sun  is  away  from  the  cen- 
ters of  the  orbits  of  Mercury  and  Mars.  On  July  4th,  the 
sun  is  in  the  constellation  Gemini.  Sagittarius  comes  to 
the  meridian  at  midnight,  and  from  the  star  maps  it  may 
be  seen  that  Aquila,  Lyra,  Draco,  etc.,  come  to  the  meridian 
at  the  same  time  with  Sagittarius,  since  they  have  the  same 
right  ascension. 

As  an  exercise  let  the  table  of  epochs  (page  43)  be  ex- 
tended two  or  three  years  beyond  1910  by  adding  to  the 
last  date  in  each  column  multiples  of  the  planet's  periodic 
time— e.  g.,  for  Venus  1910  June  28th,  -f  224.7d.  =  1911 
February  8th,  and  this  date  increased  by  2  X  224.7d.  = 
1912  May  3d,  etc.,  which  are  the  epochs  for  the  respective 
years.  Treat  the  other  planets  in  the  same  way. 

31.  Answers. — Mars  comes  into  opposition,  but  Mercury 
and  Venus  do  not.  The  maximum  angular  distance  of 
Venus  from  the  sun  (elongation)  may  be  found  from  Fig. 
17  by  measuring  the  angle  between  two  lines  drawn  from 
the  position  of  the  earth,  one  to  the  sun,  the  other  tangent 
to  the  orbit  of  Venus.  This  angle  is  approximately  45°. 
From  the  same  figure  it  is  apparent  that  the  shortest  dis- 
tance between  the  orbits  of  Mars  and  the  earth  is  on  line 
toward  the  constellation  Aquarius,  and  Fig.  16  shows,  by 
the  number  IX,  that  the  earth  is  in  this  part  of  its  orbit  at 
the  beginning  of  September,  which  is  therefore  the  month 
of  nearest  approach  and  maximum  brightness.  But  also 
note  from  the  figure  that  Mars  does  not  come  to  this  point 
of  its  orbit  in  every  September. 

From  Fig.  17,  Venus  may  approach  nearer  to  the  earth 


SUGGESTIONS  TO  TEACHERS  9 

than  any  other  planet.  The  earth  comes  to  the  same  longi- 
tude on  approximately  the  same  day  of  each  year,  because 
the  interval  between  successive  returns  has  been  adopted  as 
the  unit  of  time — a  year. 

The  intelligent  teacher  will  be  able  to  devise  for  him- 
self numerous  additional  exercises  to  be  worked  out  in  con- 
nection with  Figs.  16  and  17 — e.  g.,  When  will  Mercury  and 
Venus  be  simultaneously  visible  as  evening  stars  ?  When 
will  they  be  at  their  greatest  angular  distance  (elongation) 
from  the  sun  ?  When  will  Jupiter  and  Saturn  appear  close 
together  in  the  sky  (conjunction),  etc.? 

CHAPTEK  IV 

Much  that  is  contained  in  this  chapter  will  be  familiar 
to  pupils  who  have  studied  physics,  and  may  be  treated  as 
a  review.  The  exercises,  however,  should  be  worked  out 
in  order  to  furnish  a  clear  understanding  of  the  principles. 
In  the  presence  of  the  class  obtain  from  Fig.  16  the  peri- 
odic time  of  Jupiter,  11. 9y.,  and  from  §  134  its  mean  dis- 
tance from  the  sun,  5.2.  Cube  one  of  these  numbers, 
square  the  other,  and  show  that  the  quotient  a3  -f-  T*  is  in 
fact  very  nearly  unity.  Ask  the  class  to  work  out  the  cor- 
responding quotient  for  other  planets. 

37.  Illustrate  the  difference  between  mass  and  weight. 
A  bullet  fired  upward  from  a  gun  weighs  less  at  the  summit 
of  its  path  than  at  the  muzzle  of  the  gun,  because  farther 
away  from  the  earth,  but  its  mass  is  unchanged. 

38.  Fig.  20.     A  body  at  P  moving  in  the  opposite  direc- 
tion to  that  shown  in  the  figure  would  describe  toward  the 
right  a  continuation  of   a  conic  section  similar  to  those 
there  shown. 

40.  The  mathematical  symbols  refer  to  the  first  equa- 
tion on  page  55. 

42.  Fig.  23  is  intended  to  make  clear  the  formation  of 
a  tide  upon  that  side  of  the  earth  opposite  the  moon.  This 


10  ASTRONOMY 

is  usually  a  difficult  matter  for  the  pupil  to  understand, 
and  his  attention  should  be  especially  directed  to  the  be- 
havior of  the  blocks  numbered  1  and  3  in  the  figure.  Also 
note  in  Fig.  24  that  while  there  are  tidal  waves  on  opposite 
sides  of  the  earth,  they  do  not  stand  directly  under  the 
moon,  but  are  swept  along  for  a  certain  distance  by  the 
earth's  rotation. 


CHAPTER  V 

In  connection  with  this  chapter,  have  the  pupil  read 
Coleridge's  Ancient  Mariner  with  reference  to  its  numer- 
ous astronomical  allusions.  Trace  out  by  means  of  these 
as  nearly  as  may  be  the  path  of  the  ship. 

45.  Fig.  26  is  intended  to  illustrate  only  the  principles 
involved  in  determining  the  mass  of  the  earth.     The  actual 
application  of  these  principles  must  be  made  by  means  of 
much  more  elaborate  apparatus.     In  the  case  suggested  in 
the   text,  the   displacement  of  the  plumb  bob  would  be 
approximately  0.00001  inches,  as  may  be  seen  by  solving 
for  d  the  last  equation  on  page  73.     If  the  pupils  have  a 
knowledge  of  logarithms,  let  them  solve  by  their  aid  the 
last  two  problems  of  §  45.     Otherwise  omit  these,  as  the 
computations  are  tedious  when  made  by  the  ordinary  pro- 
cesses of  multiplication. 

46.  The   subject-matter  of  this   section   is  frequently 
called  "  precession  of  the  equinoxes,"  but,  as  here  devel- 
oped, the  stress  is  to  be  laid  upon  the  motion  of  the  earth's 
axis  ;  the  resultant  effect  upon  the  position  of  the  equinox 
is  a  subordinate  matter.     To  illustrate  the  precession,  hold 
a  coin  so  that  its  plane  makes  an  angle  of  23°  with  the  top 
of  a  table,  and  keeping  this  angle  unchanged,  turn  the  coin 
around  so  that  it  shall  face  in  succession  north,  east,  south, 
and  west.     The  table  represents  the  plane  of  the  ecliptic, 
the  coin  the  plane  of  the  earth's  equator,  a  pencil  held  per- 
pendicular to  the  coin  will  then  represent  the  axis  of  the 


SUGGESTIONS  TO  TEACHERS  H 

earth  and  its  motion  relative  to  the  ecliptic,  while  the  line 
in  which  the  plane  of  the  coin  intersects  the  plane  of  the 
table  indicates  by  its  successive  position  the  motion  of  the 
equinox. 

In  the  case  of  a  top  the  tipping  force  tends  to  bring  the 
axis  down  into  a  horizontal  position  ;  in  the  case  of  the 
earth  the  tipping  force  tends  to  straighten  the  axis  up,  per- 
pendicular to  the  ecliptic  ;  hence  the  wobble  of  the  top 
and  the  precession  of  the  earth's  axis  take  place  in  opposite 
directions,  one  with  the  spin,  the  other  opposed  to  it. 

Vega  will  be  the  pole  star  12,000  years  hence,  and  a  Dra- 
conis  was  the  pole  star  about  2700  B.  c.  After  a  lapse  of 
26,000  years  the  earth's  axis  will  have  made  one  complete 
"  wobble,"  and  will  again  point  in  the  same  direction,  and 
the  stars  will  again  have  very  nearly  the  same  right  ascen- 
sions and  declinations  as  now.  The  pole  is  now  moving 
away  from  the  Big  Dipper,  will  never  come  exactly  to 
Polaris,  but  after  a  lapse  of  12,000  years  will  be  distant 
about  48°  from  it — i.  e.,  a  little  more  than  the  diameter  of 
the  circle  along  which  it  moves. 

48.  Point  out  that  an  increase  in  the  obliquity  of  the 
ecliptic  would  cause  the  sun  to  rise  higher  in  the  sky  dur- 
ing summer  and  to  run  lower  during  the  winter  months, 
and  would  therefore  exaggerate  the  extremes  of  heat  and 
cold.  The  precession  has  little  effect  upon  climate,  but  if 
the  earth's  axis  were  directed  toward  Arcturus,  for  example, 
the  obliquity  of  the  ecliptic  would  be  changed,  with  result- 
ing marked  effects  upon  the  seasons. 

Fig.  27.  Insist  upon  the  fact  that  the  peculiar  shape  of 
the  sun's  reflected  image  shows  that  the  surface  of  the 
water  is  not  flat,  but  is  curved  in  some  way. 

Neglecting  the  effect  of  refraction,  evening  twilight 
lasts  at  the  north  pole  from  the  time  the  sun  passes  the 
autumnal  equinox  until  it  reaches  a  declination  of  18° 
south — i.  e.,  from  September  23d  to  November  13th, 


12  ASTRONOMY 


CHAPTEE   VI 

Practice  determining  the  equation  of  time  for  different 
dates  from  Fig.  30,  and  compare  the  results  with  the  "  sun 
fast  "  or  "  sun  slow  "  furnished  by  an  almanac. 

52.  The  statement  in  the  last  line  is  wrong  and  must 
be  corrected.  For  longer,  read  shorter. 

58.  Compare  the  watch  with  the  telegraph  signals  be- 
fore and  after  observing  the  sun,  and  allow  for  its  varia- 
tion between  the  comparisons. 

61,  62.  The  pupil  should  not  be  asked  to  learn  these 
formulae,  but  rather  to  use  them  with  the  book  before  him. 
The  numerical  factors  and  divisors  in  the  formula  of  §  62 
of  course  come  from  the  number  of  days  in  the  week  and 
from  the  numbers  given  in  the  leap-year  rule. 


CHAPTEE  VII 

Fig.  33.  The  new  moon,  at  Q,  would  produce  a  partial 
solar  eclipse  visible  in  the  regions  about  the  north  pole. 

64.  The  theorem  suggested  at  the  end  of  the  section  is 
true. 

65.  Query. — The  clouds  would  cut  off  much  of  the  light 
and  the  eclipse  would  be  a  dark  one.     The  moon  might 
disappear  altogether. 

68.  The  essence  of  a  Solar  Eclipse  Limit  is  that  when 
the  earth  is  too  far  away  from  the  node  the  new  moon  will 
stand  so  far  north  or  south  of  the  plane  of  the  earth's  orbit 
that  the  lunar  shadows  will  fail  to  strike  the  earth. 

?0.  Eclipse  prediction  for  Chicago,  Fig.  35.  Chicago 
lies  inside,  west  of,  the  curve  marked  Begins  at  lh.,  and 
distant  from  it  about  one  third  the  space  between  the 
curves  for  lh.  and  2h.  The  eclipse  therefore  began  about 
one  third  of  an  hour  before  one  o'clock,  Greenwich  Time — 
i.  e.,  at  6h.  40m.  A.  M.  Central  Standard  Time.  Similarly  it 


SUGGESTIONS  TO  TEACHERS  13 

ended  a  very  little  after  three  o'clock — e.  g.,  9h.  2m.  Cen- 
tral Time. 

Fig.  36.  Note  that  the  next  total  solar  eclipse  visible 
in  the  United  States  occurs  on  June  6,  1918,  and  will  be 
visible  from  Oregon  to  Georgia. 

72.  The  following  table  shows  all  the  eclipses  that  oc- 
curred during  the  years  1884-'89,  and  in  connection  with 
the  saros  it  will  suffice  for  predicting  the  eclipses  of  the 
years  1902-'07.  Have  the  pupils  predict  the  eclipses  of 
next  year,  giving  their  dates,  character,  whether  solar  or 
lunar,  partial,  total,  or  annular,  and  the  parts  of  the  earth 
in  which  they  are  visible.  Note  that  any  given  eclipse 
may  extend  over  a  much  larger  part  of  the  earth's  surface 
than  is  shown  in  the  table,  which  gives  approximately  the 
center  of  the  eclipse — e.  g.,  the  first  eclipse  of  the  table  was 
visible  from  England  to  Siberia  and  the  north  pole. 

Table  of  Eclipses 
1884 

March  27th Partial  solar Norway. 

April  9th-10th Total  lunar Pacific  Ocean. 

April  25th Partial  solar South  Atlantic  Ocean. 

October  4th Total  lunar Africa. 

October  18th Partial  solar Alaska. 

1885 

March  16th-17th Annular  solar North  America. 

March  30th. Partial  lunar Indian  Ocean. 

September  8th-9th Total  solar South  Pacific  Ocean. 

September  24th Partial  lunar East  Pacific  Ocean. 

1886 

March  5th-6th Annular  solar Pacific  Ocean. 

August  29th Total  solar Atlantic  Ocean. 

1887 

February  8th Partial  lunar Pacific  Ocean. 

February  22d-23d Annular  solar South  Pacific  Ocean. 

August  2d Partial  lunar Africa. 

August  19th Total  solar Asia, 


14:  ASTRONOMY 

1888 

January  28th-29th Total  lunar Northern  Africa. 

February  llth-12th. . .  Partial  solar South  Pacific  Ocean. 

July  9th Partial  solar Indian  Ocean. 

July  23d Total  lunar South  Pacific  Ocean. 

August  7th Partial  solar Arctic  Ocean. 

1889 

January  1st Total  solar California. 

January  17th Partial  lunar West  Indies. 

June  28th Annular  solar Madagascar. 

July  12th Partial  lunar Madagascar. 

December  22d Total  solar Atlantic  Ocean. 

To  apply  the  saros  to  the  first  of  these  eclipses,  March 
27,  1884,  we  note  that  in  the  18  years  following  this  date 
there  are  only  3  leap  years,  while  usually  there  are  either  4 
or  5.  The  length  of  the  saros  must  therefore  be  taken  as 
18  years  12^  days,  and  adding  this  to  the  given  date  we 
find  for  the  first  eclipse  of  1902  the  date  April  8th.  This 
will  be  a  partial  solar  eclipse  visible  in  about  the  latitude 
of  Norway,  but  one  third  of  the  way  around  the  earth 
toward  the  west — i.  e.,  in  British  America  and  Alaska. 

Note  how  the  eclipses  of  the  preceding  table  illustrate 
the  statement  made  at  the  end  of  §  69. 


CHAPTER  VIII 

This  chapter  is  written  from  the  standpoint  of  physical 
optics,  and  in  teaching  it  the  wave  front  instead  of  the  ray 
of  light  is  to  be  given  prominence.  Insist  upon  the  anal- 
ogy between  spherical  waves  in  the  ether  and  circular 
waves  propagated  along  the  surface  of  water.  Especial 
care  must  be  given  at  the  outset  to  obtaining  clear  ideas  of 
wave  lengths.  The  analogies  offered  by  sound  are  here 
very  advantageous.  Consult  any  good  text-book  of  physics. 

76.  The  principle  involved  in  the  case  of  the  elliptical 
mirror  is  that  the  sum  of  the  distances  from  the  two  foci  of 


SUGGESTIONS  TO  TEACHERS  15 

an  ellipse  to  any  point  on  its  circumference  is  the  same  as 
the  sum  of  the  corresponding  distances  for  any  other  point 
on  the  ellipse. 

79.  Try  making  a  telescope  with  a  reading  glass  for  ob- 
jective and  a  pin  hole  for  eyepiece,  and  note  how  it  shows 
objects  upside  down,  as  do  all  telescopes  used  for  astro- 
nomical purposes.     Be  sure  not  to  get  the  reading  glass  too 
near  the  pin  hole.     Try  also  a  second  reading  glass  or  other 
lens  for  an  eyepiece,  and  note  that  with  this  combination  a 
bit  of  paper  or  dirt  stuck  to  the  objective  can  not  be  seen 
through  the  eyepiece  simultaneously  with  a  distant  object, 
thus  illustrating  the  statement  made  near  the  end  of  §  80. 

80.  Figs.  41-44.     In  picking  out  corresponding  parts  of 
these  equatorial  mountings,  begin  with  the  polar  axis  (b) 
and  take  next  the  declination  axis  (c). 

Fig.  49.  The  spectroscope  is  here  shown  at  the  extreme 
left  of  the  figure.  The  prism  is  placed  at  the  elbow,  where 
the  axis  of  the  telescope  tube  meets  the  part  projecting 
obliquely  upward  and  to  the  left. 

The  principles  set  forth  in  §  85  and  §  88  should  be 
thoroughly  learned,  as  they  are  of  frequent  application  in 
subsequent  parts  of  the  text.  Compare  the  pictures  of  the 
stellar  spectra  shown  in  Chapter  XIII  with  the  diagram  of 
the  sun's  spectrum  (Fig.  50).  Note  that  all  of  these  are 
absorption  spectra,  while  those  shown  in  Figs.  47  and  48 
are  emission  (bright-line)  spectra. 

CHAPTEE  IX 

91.  Question  the  pupil  rather  closely  upon  this  section 
to  determine  whether  his  own  observation  corresponds  to 
the  statements  here  made.  A  foundation  for  this  should 
have  been  laid  during  the  preceding  weeks  by  directing  his 
attention  to  the  moon  and  suggesting  things  to  look  for. 

Fig.  53  will  repay  careful  study.  Measure  with  the 
protractor  the  angle  at  the  earth  between  moon  and  sun  on 


16  ASTRONOMY 

the  different  dates,  and  determine  from  this  angle  the  time 
at  which  the  moon  will  come  to  the  meridian— e.  g.,  on 
June  30th  the  angle  is  0°,  and  sun  and  moon  come  to  the 
meridian  together,  at  noon.  On  July  3d  the  angle  is  37.5° 
=  2h.  30m.,  and  the  moon  crosses  the  meridian  at  2.30  P.  M. 
Observe  also  the  relation  of  the  moon's  phase  to  the  time 
at  which  it  crosses  the  meridian,  new  moon  corresponding 
to  noon,  full  moon  to  midnight,  etc. 

94.  Other  methods  of  determining  the  mass  of  the 
moon  are  also  employed  by  astronomers,  but  the  one  here 
given  is  sufficient  for  illustration. 

98.  Librations. — Make  sure  that  the  pupils  measure  the 
positions  of  craters  or  other  markings  in  Fig.  55,  to  show 
that  there  is  a  real  libration  between  the  two  parts  of  the 
picture.  Also  make  sure  that  the  coin  experiment  (Fig. 
54)  is  actually  performed. 

In  genera],  whenever  references  are  made  to  features  of 
the  moon's  surface  shown  in  Fig.  55,  the  same  features 
may  also  be  found  in  the  plate  facing  page  150.  Make  use 
of  this  plate  in  working  out  the  exercises  contained  in  the 
remainder  of  the  chapter. 

102.  The  area  of  Lake  Superior  is  32,000  square  miles. 

104.  The  equation  at  the  top  of  page  171  may  be  derived 
from  that  on  page  170  by  expanding  the  binomial  and 
dropping  h2  as  being  insignificant  in  comparison  with  the 
much  larger  term  2Rh. 

Beware  that  the  pupil  does  not  infer  from  Fig.  61  that 
the  moon  has  an  atmosphere.  The  figure  is  intended  to 
show  effects  that  would  be  produced  if  there  were  an  at- 
mosphere, and  whose  absence  proves  that  there  is  none. 

CHAPTEE   X 

110.  The  Ode  to  Darkness  should  be  read  entire  by  the 
pupil. 

111.  The  number  333,000  here  given  as  the  reciprocal 


SUGGESTIONS  TO  TEfcMfJfRS    <  — "Vx        17 


of  the  earth's  mass  differs  from  the  329,000  of  §  40  because 
the  latter  corresponds  to  the  sum  of  the  masses  of  earth 
and  moon.  Use  logarithms  for  the  numerical  solution  of 
the  equations  in  this  section.  Endeavor  by  the  use  of  con- 
crete illustrations  to  give  the  pupil  a  good  idea  of  parallax 
as  a  means  of  measuring  distances — e.  g.,  Fig.  3,  where  the 
moon's  distance  is  determined  by  its  parallax,  as  seen  from 
N^orth  America  and  South  America. 

116.  Don't  ask  the  pupil  to  commit  to  memory  this  list 
of  elements.  It  is  given  for  illustration  only. 

119.  Figs.  66-69.  Have  the  pupil  j)lot  carefully  upon 
some  one  of  these  figures  the  position  of  the  group  of  spots 
as  shown  in  each  of  the  others.  A  smooth  curve  drawn 
through  the  positions  thus  plotted  will  show  the  path  of 
the  spot  across  the  sun's  disk.  The  position  of  the  sun's 
rotation  axis,  shown  by  the  straight  line  in  the  figure,  is 
determined  by  drawing  a  diameter  of  the  sun's  disk  per- 
pendicular to  the  path  of  the  spot. 

121.  The  dimensions  of  the  sun  spots  are  to  be  deter- 
mined from  Figs.  67  and  68  in  the  same  manner  as  the 
dimensions  of  lunar  craters  were  found. 

126.  Enlarge  upon  the  hand-pump  experiment,  and 
show  by  feeling  of  different  parts  of  the  apparatus  that  the 
heat  comes  from  compression  of  the  air,  and  not  from  ordi- 
nary mechanical  friction. 

129.  The  most  recent   determination   of   the   sun-spot 
period,  by  Newcomb,  makes  its  average  length  11.13  years, 
and  fixes  as  the  probable  epoch  of  the  next  maximum, 
December,  1904. 

130.  Draw  a  circle  of  the  size  of  Fig.  82,  and  copy  upon 
it  from  that  figure  the  lines  showing  the  position  and  fre- 
quency of  sun  spots  for  the  year  1879.     Construct  a  similar 
figure  for  each  of  the  other  years  shown  in  Fig.  82,  and 
arrange  them  in  sequence,  so  as  to  show  the  progressive 
variation  in  the  distribution  of  sun  spots, 


18  ASTRONOMY 


CHAPTEK   XI 

In  Fig.  85  E  represents  the  earth. 

Bode's  law.  —  Point  out  that  Uranus  and  the  asteroids 
fall  in  with  Bode's  law,  although  unknown  at  the  time  it 
was  first  published,  while  Xeptune's  distance  from  the  sun 
does  not  at  all  agree  with  the  law.  Ask  the  pupil  for  an 
independent  judgment  of  Bode's  law.  Is  it  probably  a  real 
law  of  Mature  or  a  chance  coincidence  ?  The  opinions  of 
astronomers  upon  this  subject  are  divided. 

The  order  in  which  the  planets  are  treated  is  quite  un- 
conventional, but  is  based  upon  the  sequence  in  stages  of 
development  presented  by  them,  Jupiter  standing  nearest 
to  the  sun,  while  the  small  planets,  Mercury  and  Mars,  are 
farthest  removed  in  respect  of  physical  condition.  It  also 
seemed  desirable  to  deal  with  Mars  immediately  before 
taking  up  the  question  of  life  upon  the  planets. 

136.  It  is  shown  in  §  148  that  the  phase  of  a  planet  de- 
pends upon  the  angle  at  the  planet  between  earth  and  sun. 
From  Fig.  16  it  is  easy  to  find  that  in  the  case  of  Jupiter 
this  angle  can  never  much  exceed  12°,  and  •££$  is  therefore 
the  greatest  amount  of  dark  area  that  Jupiter's  face  can 
ever  turn  toward  the  earth. 

142.  Have  the  pupils  determine  the  phase  of  Saturn's 
ring  for  the  date  of  the  next  opposition  —  e.  g.,  for  1902 
Saturn  will  be  in  Capricornus  (see  Fig.  16),  and  at  that 
time  (Fig.  92)  the  northern  side  of  the  ring  will  be  visible, 
but  the  ring  will  not  be  opened  out  to  its  full  extent. 

145.  The  radius  of  the  outer  edge  of  the  'ring  is  86,000 
miles,  and  by  Kepler's  Third  Law  we  have  for  the  par- 
ticles composing  it  and  for  the  satellite,  Titan,  whose  peri- 
odic time  is  383  hours,  the  proportion 


(86,000)3  _  _ 

~T*  (383) 


SUGGESTIONS  TO  TEACHERS  19 

which  when  solved  furnishes  for  the  rotation  time  of  the 
outer  edge  of  the  ring  jT=14.2h.  Determine  the  inner 
radius  of  the  ring  by  measurement  from  Fig.  93,  and  apply 
the  above  method  to  determine  its  periodic  time. 

148.  Phases  of  V7enus. — The  distance  of  Venus  from  the 
earth  is  of  course  inversely  proportional  to  its  apparent 
diameter,  and  the  distances  scaled  from  Fig.  17  should  con- 
firm this. 

In  Fig.  17  the  angle  at  Venus  between  the  earth  and 
sun  is  about  172°,  and  this  number  divided  by  180  gives  as 
a  quotient  0.955,  from  which  it  appears  that  on  July  4, 
1900,  Venus  presented  to  the  earth  a  very  narrow  crescent 
of  light,  more  than  95  per  cent  of  its  visible  hemisphere 
being  dark. 

150.  Next  favorable  opposition  of  Mars. — In  Fig.  17  try 
whether  the  planet  approaches  near  the  earth  in  the  year 
1907. 

151.  Determine  from  Fig.  17  the  season  in  the  northern 
hemisphere  of  Mars  at  the  time  of  its  next  opposition.     In 
1902  Mars  crosses  the  initial  line,  F,  on  March  16th ;  the 
earth  196  days  later,  on  September  23d.     Following  the 
numbers  on  the  orbit  of  Mars  and  the  earth,  we  find  oppo- 
site the  constellation  Virgo  a  place  where  the  numbers  for 
Mars  are  196  greater  than  for  the  earth — i.  e.,  the  place  at 
which  the  earth  will  have  caught  up  with  Mars,  and  where 
the  planet  will  appear  in  opposition.     But  Virgo  is  just 
opposite  to  Pisces,  and  the  sun  as  seen  from  Mars  will  be 
in  Pisces  at  the  opposition  of  April,  1903,  and  it  will  be 
summer  in  the  northern  hemisphere  of  Mars. 

154.  The  name  Schiaparelli  is  pronounced  Ske-ap-ar- 
el-ly. 

157.  From  Fig.  84  the  force  of  gravity  upon  Mars  is 
0.44  times  as  great  as  on  the  earth.  A  man  who  weighs 
150  pounds  here  would  weigh  66  pounds  there  ;  he  could 
jump  more  than  twice  as  far,  etc.  Have  the  pupils  work 
this  out  for  -both  Mars  and  Venus. 


20  ASTRONOMY 


CHAPTER   XII 

Give  especial  heed  to  Figs.  109  and  110  as  illustrating 
the  motion  of  the  comet.  The  angle  between  the  plane  of 
the  comet's  orbit  and  the  plane  of  the  earth's  orbit  is  38°, 
and  the  fact  that  the  two  orbits  do  not  lie  in  the  same 
plane  should  be  impressed  upon  the  pupil. 

165.  Have  the  pupils  watch  for  meteors  on  some  suit- 
able night,  and  count  the  number  that  can  be  seen  per 
hour.  If  several  persons  can  work  together,  each  keeping 
watch  upon  a  limited  part  of  the  sky,  the  results  will  be 
more  satisfactory.  Impress  upon  the  pupil  that  a  meteor 
is  not  very  different  from  an  ordinary  stone,  and  that  a 
dozen  average  meteors  might  be  carried  in  one's  pocket 
without  inconvenience.  But  some  meteors  are  very  much 
too  large  for  this,  and  weigh  hundreds  if  not  thousands  of 
pounds. 

170.  Construct  a  figure  showing  the  sun,  the  earth,  and 
the  direction  of  the  earth's  orbital  motion  and  rotation 
about  its  axis.  It  will  be  apparent  from  this  figure  that 
the  part  of  the  earth  at  which  it  is  morning  is  on  the 
front  side. 

Fig.  113.  Note  that  the  straight  lines  which  represent 
the  meteor  paths  are  in  part  heavy,  corresponding  to  the 
time  in  which  the  meteor  was  incandescent,  and  in  part 
light,  corresponding  to  the  time  before  and  after  its  visi- 
bility. The  chief  purpose  of  the  cut  is  to  show  how  these 
parallel  paths,  as  projected  back  against  the  sky,  appear  to 
radiate  from  the  point  a.  This  radiant  shares  in  the  diur- 
nal motion  of  the  sky.  No  meteor  which  belongs  to  the 
shower  in  question  can  cross  the  radiant.  The  radiant  is 
not  strictly  a  point,  but  a  small  area — e.  g.,  in  some  cases 
twice  as  big  as  the  full  moon — showing  that  the  paths  of 
the  meteors  are  not  strictly  parallel,  although  nearly  so. 


SUGGESTIONS  TO  TEACHERS  21 

Fig.  115.  The  earth  encounters  these  meteor  showers 
about  the  middle  of  August  and  November  respectively ; 
the  latter  is  the  Leonid  shower,  and  the  former  have  their 
radiant  in  the  constellation  Perseus,  and  are  called  Per- 
seids.  Their  radiant  points  are  north  of  the  ecliptic 
(above  in  the  figure),  and  the  meteors  are  best  seen  in 
the  morning  hours,  although  some  of  them  are  visible  in 
the  evening. 

177.  Any  comet  moving  in  a  parabola  would  require  195 
years,  (150)^  -=-  (89)*,  to  recede  from  the  sun  to  a  distance 
150  times  as  great  as  that  of  the  earth. 

178.  See   Clerke,  History  of  Astronomy  in  the  Nine- 
teenth Century,  for  an  account  of  Encke's  Comet  and  its 
impeded  motion. 


CHAPTER  XIII 

186.  The  exercises  at  the  end  of  this  section  are  to  be 
solved  by  means  of  the  equation  there  given.     Thus  to 
compare  Spica  with  a  sixth-magnitude  star,  m  =  l,  n  =  6, 

6-1 

B  —  100  5  "=  100 — i.  e.,  the   one   star  is   a  hundred-fold 
brighter  than  the  other. 

The  number  of  seventeenth  magnitude  stars  required  to 
make  one  of  the  sixth  magnitude  is 

100^  =1002  X  100*  =  100s  X  2.5119  =  25,119. 
The  required  number  of  full  moons  is 

10026'«"12  =  1002-9  =  630,100 

— i.  e.,  more  than  three  times  as  many  as  could  be  packed 
into  the  whole  visible  sky  from  zenith  to  horizon. 

187.  Assuming  the  stars  whose  magnitudes  lie  between 
5.0  and  6.0  to  have  on  the  average  the  magnitude  5.5,  we 


22  ASTRONOMY 

5.6-1 

find  from  the  equation  B  =  100  5  that  a  first-magnitude 
star  gives  as  much  light  as  63  such  stars.  It  would  there- 
fore require  45  first-magnitude  stars  to  give  as  much  light 
as  the  2,843  stars  in  question — i.  e.,  these  faint  stars  col- 
lectively give  more  light  than  do  all  the  39  stars  brighter 
than  the  second  magnitude. 

By  an  entirely  similar  process  we  find  for  the  number 
of  tenth-magnitude  stars  required  to  give  the  same  amount 
of  light,  a  little  more  than  179,000. 

188.  The  number  206,265  is  the  number  of  seconds  con- 
tained in  an  arc  equal  in  length  to  the  radius  of  the  circle 
to  which  the  arc  belongs. 

189.  Find  on  the  star  maps  a  few  of  the  stars  shown  in 
Fig.  122,  using  for  this  purpose  the  right  ascensions  and 
declinations  given  in-  the  table — e.  g.,  show  in  this  way  that 
two  such  stars  as  Nos.  33  and  34,  which  appear  near  each 
other  in  Fig.  122,  are  really  widely  'separated  in  the  sky. 
Their  apparent  proximity  in  the  figure  comes  fr*m  the 
unavoidable  neglect  of  their  declinations  in  its  construc- 
tion. 

191.  Insist  that  the  pupil  shall  trace  out  upon  the  sky 
some  of  the  allineations  given  in  this  section. 

192.  In  Fig.   113  the  apparent  proper  motions  of  the 
meteors  are  shown  by  the  herring-bone  markings  upon  the 
semicircle  which  represents  the  sky. 

193.  Show  by  a  graphical  construction  like  that  of  §  7 
how  the  angular  proper  motion  of  a  star,  given  in  seconds 
per  year,  can  be  turned  into  linear  measure,  miles  per  year, 
when  the  distance  is  known.     The   proper  motion  corre- 
sponds to  the  angle  subtended  by  the  window. 

194.  The  question  at  the  end  of  this  section,  of  course, 
does  not  admit  of  a  direct  answer.     It  is  inserted  as  an  ap- 
peal to  the  pupil's  imagination. 

196.  Putting  m  =  6  in  the  equation  for  solar  stars,  we 
find  D  =  23  X  23  =  184  years,  which  is  the  time  required 
by  light  to  come  from  an  average  sixth-magnitude  star  of 


SUGGESTIONS  TO  TEACHERS  23 

the  solar  type.     For  an  average  Sirian  star  of  the  sixth 
magnitude  D  —  416  years. 

197.  The  average  distance  of  Jupiter  from  the  earth  is 
5.2  ;  the  distance  of  a  Centauri  (§  189)  is  0.27  X  1,000,000 
=  270,000,  and  the  ratio  of  these  numbers  is  easily  found 
by  means  of  logarithms  to  be  104'72,  or  51,880 — i.  e.,  a  Cen- 
tauri is  51,880  times  as  remote  as  Jupiter.  But  since  each 
tenfold  increase  of  distance  changes  the  brightness  by  five 
magnitudes,  Jupiter's  brilliancy  at  the  distance  of  a  Cen- 
tauri would  be  4.72  X  5  =  23.6  magnitudes,  less  than  at 
present,  and  its  stellar  magnitude  would  therefore  be  —1.7 
+  23.6  =  21.9. 

200.  The  periodic  times  in  the  table  are  expressed  in 
years. 

201.  The  relative  brightness  of  Sirius  and  its  companion 
is   found  by  applying  the  equation  of  §  186.     For  Sirius 
B  =  10,000,  and  for  Procyon  B  =  25,000. 

203.  The  problem  in  this  section  is  readily  solved  if  we 
note  that  the  periodic  time  in  seconds  multiplied  by  the 
velocity  is  equal  to  the  circumference  of  the  orbit. 

Xote  that  one  of  the  components  of  a  spectroscopic 
binary  may  be  a  dark  star,  and  that  in  the  case  of  the  stars 
Spica,  Algol,  and  others,  this  is  actually  the  case.  The 
binary  nature  of  the  star  is  then  shown  by  the  variable 
velocity  of  its  bright  component  in  -the  line  of  sight,  in- 
stead of  by  a  doubling  of  the  lines.  See  Fig.  130. 

204.  Have  the  pupil  look  up  Algol  and  Mira  in  the  sky, 
and  observe  their  change  of  brightness.     This  may  require 
some  considerable  diligence  and  patience,  but  ultimate  suc- 
cess may  be  expected. 

205.  The  43  days  specified  in  this  section  is  very  nearly 
a  multiple  of  Algol's  periodic  time,  and  its  magnitude  at 
the  end  of  43  days  will  therefore  be  very  nearly  the  same 
as  at  the  beginning. 

206.  /?  Lyrge  will  have  a  maximum  of  brilliancy  when 
the  two  components  are  so  placed,  relative  to  the  earth, 


24  ASTRONOMY 

that  neither  interferes  with  the  light  of  the  other — i.  e., 
when  they  have  turned  in  their  orbital  motion  90°  around 
from  the  position  shown  in  the  figure. 

208.  The  temporary  star  of  1901,  Nova  Persei,  although 
not  so  bright  as  that  of  Tycho  Brahe,  exceeded  in  brilliancy 
any  new  star  during  the  preceding  three  centuries.  Dis- 
covered on  February  21,  1901,  as  a  second-magnitude  star, 
it  rapidly  increased  in  brilliancy,  and  on  February  24th 
was  brighter  than  Capella — i.  e.,  it  had  a  negative  stellar 
magnitude,  — 0.5.  After  February  24th  it  faded  for  a  time 
with  unusual  rapidity,  losing  more  than  95  per  cent  of  its 
light  within  a  fortnight. 


CHAPTER  XIV 

211.  The  essential  fact  to  be  insisted  upon  here  is  that 
the  spectroscope  shows  differences  in  the  kind  of  light 
which  different  stars  emit,  and  by  means  of  these  differ- 
ences the  stars  are  classified  into  three  groups,  which  are 
not  sharply  marked  off  but  blend  into  each  other,  just  as 
men  may  be  classified  with  respect  to  stature  into  tall,  me- 
dium, and  short.  There  is  a  real  difference  with  no  sharp 
dividing  line  between  the  classes. 

213.  The  cuts  in  £his  and  the  following  sections  should 
receive  especial  attention.  Trace  out  in  them  the  features 
to  which  attention  is  directed  in  the  text. 

217.  If  we  had  two  Orion  nebulae  of  exactly  the  same 
appearance  but  one  twice  as  far  off  as  the  other,  the  second 
one  must  be  twice  as  long  and  twice  as  wide  in  order  to 
look  equally  large — i.  e.,  its  volume  must  be  four  times  as 
great,  but,  owing  to  its  double  distance,  each  unit  of  mass 
would  attract  only  one  fourth  as  strongly  as  a  similar  unit 
of  the  first  nebula.  Hence  the  total  attraction  of  the  two 
nebulae,  for  the  sun  would  be  equal  if  their  thickness  and 
density  were  equal. 


SUGGESTIONS  TO  TEACHERS  25 

218.  Figs.  148-150.     Point  out  that  each  tiny  fleck  of 
light   in    these    pictures   is   a   separate   and   independent 
star. 

219.  If  possible  obtain  a  field  glass,  and  have  each  mem- 
ber of  the  class  examine  the  Milky  Way  with  it.     Compare 
its  brighter  and  fainter  parts. 

220.  The  questions  at  the  beginning  of  this  section  are 
intended  as  a  mental  stimulus,  and  of  course  do  not  admit 
of  direct  answers. 

222.  Note  that  if  the  sun  did  not  lie  inside  the  galac- 
tic stratum  the  Milky  Way  would  be  a  small  circle  of  the 
sky. 

225.  That  starlight  is  probably  absorbed  by  dust  and 
meteoric  particles  which  fill  the  universe,  is  an  important 
concept  which  should  be  emphasized,  and  the  pupil  made 
to  realize  the  futility  of  all  attempts  to  explore  the  whole 
universe. 

CHAPTEK  XV 

A  proper  mental  attitude  toward  this  chapter  is  very 
important,  and  it  must  be  the  teacher's  task  to  secure  that 
attitude  in  the  pupil.  The  subject-matter  is  inevitably 
thrust  upon  every  philosophic  mind  that  studies  the  prob- 
lems of  astronomy,  but  the  best  results  which  can  be  at- 
tained rest  upon  reasoning  that  falls  far  short  of  demon- 
stration. On  the  other  hand,  the  views  here  presented, 
although  speculative,  are  not  to  be  classed  with  the  myths 
about  the  origin  of  things  that  fill  so  large  a  place  in  folk- 
lore, or  with  the  idle  tales  about  the  "  end  of  the  world  " 
issued  by  modern  sensational  writers.  The  chapter  is  an 
attempt  to  represent  the  best  tendencies  of  modern  thought 
upon  great  problems  that  have  found  only  imperfect  solu- 
tion, but  which  may  be  made  valuable  in  education,  not  so 
much  from  the  information  they  impart  as  from  their 
stimulus  to  the  imagination  and  the  reasoning  faculty. 


26  ASTRONOMY 

Whenever  possible,  call  out  criticism  upon  the  reasoning 
employed,  and  always  seek  to  give  the  pupil  a  mental  pic- 
ture of  the  things  discussed.  Don't  allow  the  recitation  to 
degenerate  into  a  mere  rehearsal  of  the  text.  For  collateral 
reading  upon  this  topic  consult  the  works  of  Ball,  Thom- 
son, and  Newcomb,  named  in  the  Bibliography,  page  384. 


A  Table  of  Logarithms 


No. 

Log. 

No. 

Log. 

No. 

Log. 

10 

1  000 

40 

1.602 

70 

1.845 

11 

.041 

41 

.613 

71 

.851 

12 

.079 

42 

.623 

72 

.857 

13 

.114 

43 

.633 

73 

.863 

14 

.146 

44 

.643 

74 

.869 

15 

1.176 

45 

1.653 

75 

1.875 

16 

.204 

46 

.663 

76 

.881 

17 

.230 

47 

.672 

77 

.886 

18 

.255 

48 

.681 

78 

.892 

19 

.279 

49 

.690 

79 

.898 

20 

1.301 

50 

1.699 

80 

1.903 

21 

.322 

51 

.708 

81 

.908 

22 

.342 

52 

.716 

82 

.914 

23. 

.362 

53 

.724 

83 

.919 

24 

.380 

54 

.732 

84 

.924 

25 

1.398 

55 

1.740 

85 

1.929 

26 

.415 

56 

.748 

86 

.934 

27 

.431 

57 

.756 

87 

.940 

28 

.447 

58 

.763 

88 

.944 

29 

.462 

59 

.771 

89 

.949 

30 

1.477 

60 

1.778 

90 

1.954 

31 

.491 

61 

.785 

91 

.959 

32 

.505 

62 

.792 

92 

.964 

33 

.519 

63 

.799 

93 

.968 

34 

.531 

64 

.806 

94 

.973 

35 

1.544 

65 

1.813 

95 

1.978 

36 

.556 

66 

.820 

96 

.982 

37 

.568 

67 

.826 

97 

.987 

38 

.580 

68 

.833 

98 

.991 

39 

.591 

69 

.839 

99 

.996 

40 

1.602 

70 

1.845 

100 

2.000 

OF  THE 

UNIVERSITY 
C 


27 


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